Author Archives: Dr Christian P. H. Salas

In a series of papers from 1905, his `annus mirabilis’, Albert Einstein analysed Brownian motion and derived the following partial differential equation (known as the diffusion equation) from physical principles to describe the process: This equation has as a solution the probability density function (see Einstein, A., 1956, Investigations on the Theory of Brownian Movement,Continue reading “A variant of Einstein’s partial differential equation for Brownian motion”

Chebyshev’s inequality and the Borel-Cantelli lemma are seemingly disparate results from probability theory but they combine beautifully in demonstrating a curious property of Brownian motion: that it has finite quadratic variation even though it has unbounded linear variation. Not only do the proofs of Chebyshev’s inequality and the Borel-Cantelli lemma have some interesting features themselves,Continue reading “Applying Chebyshev’s inequality and the Borel-Cantelli lemma to Brownian motion”

The literature involving fractional Brownian motion has expanded over the past three decades or so. Fractional Brownian motion actually has a long history, having been first introduced in the 1940s by the great Andrey Kolmogorov, and then reintroduced and further developed in a seminal paper by Mandelbrot and Van Ness in 1968 (for an interestingContinue reading “Brownian motion and fractional Brownian motion as self-similar stochastic processes”

Gauss’s flux theorem for gravity (also known as Gauss’s law for gravity) in differential form says that where is the gravitational force, is the Newtonian gravitational constant and is the (differential) mass density. The link with the scalar potential through gives us (a well known type of partial differential equation known as Poisson’s equation). InContinue reading “Gauss’s flux theorem for gravity, Poisson’s equation and Newton’s law”

Years ago, I was thinking about various kinds of mappings of prime numbers and wondered in particular what prime numbers would look like when projected from the (extended) real line to the -sphere by a homeomorphism linking these two spaces. When I did the calculations, I was amazed to find that prime numbers are mappedContinue reading “Pythagorean triples in a homeomorphism between the 1-sphere and the extended real line”

In this short note I want to quickly set out the mathematical details of how the Riemann sphere arises when the point at infinity is added to the complex plane to give the extended complex plane , i.e., an explicit homeomorphism which establishes the topological equivalence of the -sphere and the extended complex plane, givingContinue reading “Topological equivalence of the 2-sphere and the extended complex plane (the `Riemann sphere’)”

In this note I want to explore some of the details involved in the close analogy between: A). the way Cantor constructed the real number field as the completion of the rationals using Cauchy sequences with the usual Euclidean metric; and B). the way the p-adic number field can be similarly constructed as the completionContinue reading “The p-adic number field as a completion of the rationals”

One of the more complicated-looking Schrödinger wavefunctions arises from a scattering (i.e., positive energy) problem involving an Eckart potential. These wavefunctions are expressed in terms of Gauss hypergeometric functions and as part of some numerical work I was doing using Maple software I wanted to see how easy or difficult it would be to writeContinue reading “Maple code for quantum scattering from an Eckart potential”

Sturm-Liouville theory was developed in the 19th century in the context of solving differential equations. When one studies it in depth for the first time, however, one experiences a sudden realisation that this is the mathematics providing the basic framework for quantum mechanics. In quantum mechanics we envisage a quantum state (a time-dependent function) expressedContinue reading “Overview of Sturm-Liouville theory: the maths behind quantum mechanics”